% i
clc;
clear;
tspan = [0,10];      % Visualise solution from 0 to 5 seconds
x0 = [1,1];             % Initial condition of 1

L = 3;
kappa = 2;
rho = L + kappa/sqrt(2);
sigma = @(x,z) x^2 + 4*x - 0.5*z;
v = @(x,z) -rho*sign(sigma(x,z));
u = @(x,z) v(x,z) - 2*x^3 - 10*x^2 - 12*x + x*z + 2*z;

[t,x] = ode45(@(t,x) f_1(x,t,u),tspan,x0);

x_surface = linspace(0,2,10);
z_surface = 2*x_surface.^2 + 8*x_surface;

sigma_val = zeros(length(x),1);
u_val = zeros(length(x),1);
for i = 1:length(x)
    sigma_val(i) = sigma(x(i,1),x(i,2));
    u_val(i) = u(x(i,1),x(i,2));
end

% Visualise solutions
figure('Name','i_1','NumberTitle','off','Position', [100, 100, 700, 400]);
plot(x(:,1),x(:,2),'Linewidth',2);
hold on
plot(x_surface,z_surface,'Linewidth',2);
title('x(t), z(t) With sliding surface \sigma = 0');
xlabel('x(t)');
ylabel('z(t)');

figure('Name','i_2','NumberTitle','off','Position', [100, 100, 700, 400]);
subplot(1,3,1);
plot(t,x(:,1),'Linewidth',2);
title('t, x(t)');
xlabel('Time (s)');
ylabel('x(t)');
xlim(tspan);

subplot(1,3,2);
plot(t,x(:,2),'Linewidth',2);
title('t, z(t)');
xlabel('Time (s)');
ylabel('z(t)');
xlim(tspan);

subplot(1,3,3);
plot(t,sigma_val,'Linewidth',2);
title('t, \sigma(t)');
xlabel('Time (s)');
ylabel('\sigma(t)');
xlim(tspan);

figure('Name','i_3','NumberTitle','off','Position', [100, 100, 700, 400]);
plot(t,u_val,'Linewidth',2);
title('t, u_{sm}(t)');
xlabel('Time (s)');
ylabel('u_{sm}(t)');
xlim(tspan);

%% ii
clear x z
epsilon = 0.1;
v = @(x,z) -rho*min(max((sigma(x,z)/epsilon),-1),1);
u = @(x,z) v(x,z) - 2*x^3 - 10*x^2 - 12*x + x*z + 2*z;
[t,x] = ode45(@(t,x) f_1(x,t,u),tspan,x0);
sigma_val = zeros(length(x),1);
u_val = zeros(length(x),1);
for i = 1:length(x)
    sigma_val(i) = sigma(x(i,1),x(i,2));
    u_val(i) = u(x(i,1),x(i,2));
end

% Visualise solutions
figure('Name','ii_1','NumberTitle','off','Position', [100, 100, 700, 400]);
subplot(1,3,1);
plot(t,x(:,1),'Linewidth',2);
title('t, x(t)');
xlabel('Time (s)');
ylabel('x(t)');
xlim(tspan);

subplot(1,3,2);
plot(t,x(:,2),'Linewidth',2);
title('t, z(t)');
xlabel('Time (s)');
ylabel('z(t)');
xlim(tspan);

subplot(1,3,3);
plot(t,sigma_val,'Linewidth',2);
title('t, \sigma(t)');
xlabel('Time (s)');
ylabel('\sigma(t)');
xlim(tspan);

figure('Name','ii_2','NumberTitle','off','Position', [100, 100, 700, 400]);
plot(t,u_val,'Linewidth',2);
title('t, u_\epsilon(t)');
xlabel('Time (s)');
ylabel('u_\epsilon(t)');
xlim(tspan);

%%
function dX = f_1(x,t,u)
dx = x(1)^2 + 3*x(1) - 0.5*x(2);
dz = -2*u(x(1),x(2)) + cos(t) + 2*sin(3*x(1)*x(2));
dX = [dx;dz];
end